Does the presence of intrusive thoughts and strange ideas during prayer and throughout the day mean that the person afflicted by them is not among the sincere worshippers, based on the verses that indicate that Satan cannot overpower the sincere?

The aforementioned verse does not mean that Satan's whispers (waswasa) occur only to those who are not Allah's devoted and sincere servants. Rather, Satan targets with his whispers anyone who turns to Allah the Almighty and approaches Him through His remembrance or other acts of obedience.